# Markov property

*for a real-valued stochastic process , *

The property that for any set of times from and any Borel set ,

(*) |

with probability 1, that is, the conditional probability distribution of given coincides (almost certainly) with the conditional distribution of given . This can be interpreted as independence of the "future" and the "past" given the fixed "present" . Stochastic processes satisfying the property (*) are called Markov processes (cf. Markov process). The Markov property has (under certain additional assumptions) a stronger version, known as the "strong Markov property" . In discrete time the strong Markov property, which is always true for (Markov) sequences satisfying (*), means that for each stopping time (relative to the family of -algebras , ), with probability one

#### References

[1] | I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 2 , Springer (1975) (Translated from Russian) |

#### Comments

#### References

[a1] | K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960) |

[a2] | J.L. Doob, "Stochastic processes" , Wiley (1953) |

[a3] | E.B. Dynkin, "Markov processes" , 1 , Springer (1965) (Translated from Russian) |

[a4] | T.G. Kurtz, "Markov processes" , Wiley (1986) |

[a5] | W. Feller, "An introduction to probability theory and its applications" , 1–2 , Wiley (1966) |

[a6] | P. Lévy, "Processus stochastiques et mouvement Brownien" , Gauthier-Villars (1965) |

[a7] | M. Loève, "Probability theory" , II , Springer (1978) |

**How to Cite This Entry:**

Markov property.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Markov_property&oldid=11571